Working Paper The output gap, the labor wedge, and the dynamic behavior of hours

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1 econstor Der Open-Access-Publikationsserver der ZBW Leibniz-Informationszentrum Wirtschaft The Open Access Publication Server of the ZBW Leibniz Information Centre for Economics Sala, Luca; Söderström, Ulf; Trigari, Antonella Working Paper The output gap, the labor wedge, and the dynamic behavior of hours Sveriges Riksbank Working Paper Series, No. 246 Provided in Cooperation with: Central Bank of Sweden, Stockholm Suggested Citation: Sala, Luca; Söderström, Ulf; Trigari, Antonella (21) : The output gap, the labor wedge, and the dynamic behavior of hours, Sveriges Riksbank Working Paper Series, No. 246 This Version is available at: Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. zbw Leibniz-Informationszentrum Wirtschaft Leibniz Information Centre for Economics

2 SVERIGES RIKSBANK 246 WORKING PAPER SERIES The Output Gap, the Labor Wedge, and the Dynamic Behavior of Hours Luca Sala, Ulf Söderström and Antonella Trigari SEPTEMBER 21

3 WORKING PAPERS ARE OBTAINABLE FROM Sveriges Riksbank Information Riksbank SE Stockholm Fax international: Telephone international: info@riksbank.se The Working Paper series presents reports on matters in the sphere of activities of the Riksbank that are considered to be of interest to a wider public. The papers are to be regarded as reports on ongoing studies and the authors will be pleased to receive comments. The views expressed in Working Papers are solely the responsibility of the authors and should not to be interpreted as reflecting the views of the Executive Board of Sveriges Riksbank.

4 The Output Gap, the Labor Wedge, and the Dynamic Behavior of Hours Luca Sala Ulf Söderström Antonella Trigari Sveriges Riksbank Working Paper Series No. 246 September 21 Abstract We use a standard quantitative business cycle model with nominal price and wage rigidities to estimate two measures of economic inefficiency in recent U.S. data: the output gap the gap between the actual and efficient levels of output and the labor wedge the wedge between households marginal rate of substitution and firms marginal product of labor. We establish three results. (i) The output gap and the labor wedge are closely related, suggesting that most inefficiencies in output are due to the inefficient allocation of labor. (ii) The estimates are sensitive to the structural interpretation of shocks to the labor market, which is ambiguous in the model. (iii) Movements in hours worked are essentially exogenous, directly driven by labor market shocks, whereas wage rigidities generate a markup of the real wage over the marginal rate of substitution that is acyclical. We conclude that the model fails in two important respects: it does not give clear guidance concerning the efficiency of business cycle fluctuations, and it provides an unsatisfactory explanation of labor market and business cycle dynamics. Keywords: Business cycles, Efficiency, Labor markets, Monetary policy. JEL Classification: E32, E24, E52. Sala and Trigari: Department of Economics and IGIER, Università Bocconi, Milan, Italy, luca.sala@unibocconi.it, antonella.trigari@unibocconi.it; Söderström: Research Division, Monetary Policy Department, Sveriges Riksbank, Stockholm, Sweden, and CEPR, ulf.soderstrom@riksbank.se. We have benefited from discussions with and comments from Mikael Carlsson, Ferre De Graeve, Simona Delle Chiaie, Michael Evers, Andrea Ferrero, Jordi Galí, Tommaso Monacelli, Giorgio Primiceri, Stephanie Schmitt-Grohé, Lars Svensson, Anders Vredin, Karl Walentin, and Andreas Westermark. We also thank seminar participants at Sveriges Riksbank, Norges Bank, the Norwegian School of Management, Oxford University, University College London, ESSIM 29, the 8th Macroeconomic Policy Research Workshop on DSGE Models at Magyar Nemzeti Bank in September 29, the Royal Economic Society 21 Annual Conference, the Konstanz Seminar on Monetary Theory and Policy 21, and the Ensuring Economic and Employment Stability conference on Monetary and Fiscal Policy for Macroeconomic Stability in June 21. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting the views of the Executive Board of Sveriges Riksbank.

5 1 Introduction A classic question in macroeconomics concerns the extent to which business cycle fluctuations are efficient. Different schools of thought have provided very different answers to this question. Traditional Keynesian theory implied that business cycle fluctuations are mainly inefficient and therefore should be counteracted by economic policy (Modigliani (1977)). In contrast, real business cycle theory suggested that most fluctuations may be driven by the efficient responses of firms and households to exogenous shifts in technology and preferences, reducing the role for countercyclical economic policy (see, for instance, Kydland and Prescott (1982), Long and Plosser (1983), or King, Plosser, and Rebelo (1988)). Modern monetary business cycle models starting with Yun (1996), Goodfriend and King (1997), and Rotemberg and Woodford (1997) introduce Keynesian features such as imperfect competition and nominal rigidities into the real business cycle framework. Recent developments have demonstrated that quantitative versions of these models are competitive with statistical reduced-form models in fitting and forecasting the behavior of aggregate macroeconomic variables (see, for instance, Smets and Wouters (27)). This class of models is therefore potentially useful to estimate the degree to which business cycle fluctuations are efficient. In this paper, we use a standard quantitative business cycle model with nominal price and wage rigidities to estimate two measures of economic inefficiency in recent U.S. data: the output gap and the labor wedge. The output gap is the deviation of actual output from its potential level, defined as the level of output with imperfect competition but in the absence of nominal rigidities. By construction, the potential level of output is at a constant distance from the efficient level, which is the level of output with perfect competition. Variations in the efficient and potential levels of output reflect the neoclassical (or RBC) features of the economy, whereas the output gap reflects the Keynesian features. The output gap thus measures the inefficient fluctuations in output. 1 The output gap is also an important indicator for monetary policy: it is typically one of the arguments in the welfare-based loss function that is relevant for optimal monetary policy (Woodford (23)), and it is often used in descriptive models of monetary policy, such as Taylor rules. Central banks therefore monitor various output gap estimates, and much work has been aimed at estimating potential output and the output gap in this class of models. 2 The labor wedge is instead a measure of inefficiency in the allocation of labor. It is defined as the deviation of households marginal rate of substitution between consumption and leisure from firms marginal product of labor. 3 According to the Lucas and Rapping (1969) theory 1 The distance between the efficient and potential levels of output is determined by the average price and wage markups that result from imperfect competition in goods and labor markets. These markups are zero in the efficient allocation but positive (and constant) in the potential allocation. The actual level of output is also affected by exogenous shocks to the two markups as well as endogenous movements in the markups due to price and wage rigidities. 2 See, for instance, Neiss and Nelson (23), (25), Edge, Kiley, and Laforte (28), Sala, Söderström, and Trigari (28), Justiniano and Primiceri (28), Basu and Fernald (29), or Coenen, Smets, and Vetlov (29). Kiley (21) gives an overview of different definitions and uses of potential output and the output gap. 3 Chari, Kehoe, and McGrattan (27) were the first to use the term labor wedge. Other authors have 1

6 of intertemporal substitution in labor supply, the efficient allocation of labor is achieved when the marginal rate of substitution and the marginal product of labor are equalized (and are equal to the real wage). Any deviations from this efficiency condition therefore lead to inefficiencies in the allocation of the labor input. Our theoretical model implies a relationship between the output gap and the labor wedge. The two measures are exactly proportional in a simple version of the model, but in our larger model the relationship is more involved. We will show, however, that the output gap and the labor wedge are closely related also in our quantitative model, suggesting that most inefficiencies in output are due to the inefficient allocation of labor. As a consequence, we can gain insights into the sources of output gap fluctuations by studying the determinants of the labor wedge. We also find that the estimates of the output gap and the labor wedge are sensitive to the structural interpretation of labor market shocks. In this class of models, shocks to the disutility of supplying labor are observationally equivalent to shocks to the markup of the real wage over households marginal rate of substitution. But these two shocks have different implications in terms of efficiency: labor disutility shocks affect the efficient allocation and therefore lead to efficient movements in labor supply and output, whereas wage markup shocks move the actual allocation relative to the efficient allocation and thus generate inefficient fluctuations in labor and output. The interpretation of the estimated labor market shocks therefore has important consequences for the estimated output gap and labor wedge. We find that the gap and the wedge are strongly procyclical when persistent labor market shocks are interpreted as shocks to the wage markup, but essentially acyclical when the shocks are interpreted as labor disutility shocks. In order to focus on the fundamental driving forces of business cycle fluctuations, as opposed to the efficient or inefficient nature of those fluctuations, we define a variant of the labor wedge, which in the context of our model we call the fundamental labor wedge. The fundamental wedge is closely related to the labor wedge studied in the literature and is independent of the structural interpretation of the shocks. We find that the fundamental wedge is essentially driven by movements in the labor input, hours worked. We then study the determinants of the fundamental wedge and use it to interpret movements in hours, the labor wedge, and the output gap. In principle, fluctuations in the fundamental wedge can be due to endogenous movements in price and wage markups, exogenous shocks to price and wage markups, or exogenous shocks to household preferences. Galí, Gertler, and López- Salido (27) and Shimer (29) discuss which of these explanations (endogenous markups or exogenous shocks) is most plausible. Their theoretical analysis provides only indirect evidence as to the driving forces of the labor wedge; our quantitative model instead gives more precise answers. We find that movements in the fundamental wedge and hours worked are largely exogenous in our model, and are directly driven by persistent labor market shocks. The endogenous component of the wedge, given by movements in the wage markup generated by wage rigidities, is essentially acyclical. In one case when the persistent labor market shock is interpreted as also studied the labor wedge, for instance, Hall (1997), Galí, Gertler, and López-Salido (27), and Shimer (29). See Shimer (29) for additional references. 2

7 a wage markup shock the total wage markup is countercyclical, but this is entirely due to exogenous markup shocks. The model therefore does not provide a satisfactory explanation of the joint dynamics of hours and wages: fluctuations in hours are essentially exogenous, and the endogenous markup of the real wage over the marginal rate of substitution (the component of the markup that is due to wage rigidities) is acyclical. Whether fluctuations in the fundamental wedge and hours are efficient or inefficient depends on the interpretation of the labor market shocks. With persistent wage markup shocks, most fluctuations in hours are inefficient, and generate movements in the output gap and the labor wedge. With persistent labor supply shocks, in contrast, movements in hours are largely efficient, and only the remaining inefficient fluctuations are reflected in the output gap and the labor wedge. Our model is unable to distinguish between these two shocks. More broadly, however, our results suggest that understanding the sources of fluctuations in hours is essential when interpreting movements in output. We conclude that the model fails in two important respects. First, as the interpretation of labor market shocks is ambiguous, the model does not give clear guidance concerning the efficiency of business cycle fluctuations. Depending on the interpretation of these shocks, most fluctuations in output are either efficient or inefficient. Second, the model provides an unsatisfactory explanation of labor market and business cycle dynamics. Fluctuations in hours worked over the business cycle are directly due to exogenous shocks to the labor market, rather than the endogenous propagation of all shocks in the economy. The failure of the neoclassical model to reconcile the behavior of hours and the real wage is an old issue in macroeconomics, going back at least to Hall (198), Altonji (1982), and Mankiw, Rotemberg, and Summers (1985). Our results show that the issue is not resolved in modern business cycle models, despite the presence of imperfect competition and nominal rigidities. This finding casts doubt on the usefulness of this class of models to study the dynamics of labor markets and business cycles. The ambiguous interpretation of labor market shocks is another drawback of the standard framework, as highlighted by Chari, Kehoe, and McGrattan (29). Galí, Smets, and Wouters (21) approach this identification problem by reinterpreting the model such that the rate of unemployment is proportional to the wage markup. They then use data on unemployment to estimate the model and identify the two shocks. Their approach solves the identification problem, but most features of the model remain unaltered. Labor market dynamics therefore has to be largely exogenous also in their estimated model. We conclude that a different class of models is needed to make further progress in our understanding of labor market and business cycle dynamics. One promising route involves models with search and matching frictions in the labor market and nominal price and wage rigidities, as in Gertler, Sala, and Trigari (28). The paper is organized as follows. We begin by presenting our model framework in Section 2. In Section 3 we show how the output gap and the labor wedge are related in the theoretical model, and we define the fundamental wedge that we use to interpret business cycle fluctuations. We then move on to estimate the model in Section 4, and Section 5 provides our main results. We examine the robustness of our results by analyzing some alternative specifications of the empirical model in Section 6. Finally, in Section 7 we offer 3

8 some concluding remarks and we point to possible directions for future research. An Appendix provides details about the model and data used. 2 The model economy Our model is a monetary Dynamic Stochastic General Equilibrium (DSGE) framework, and is similar to many models used in the literature. The particular specification we use builds closely on Smets and Wouters (27) and Justiniano, Primiceri, and Tambalotti (21). The model combines a real business cycle core with Keynesian features. The core RBC model features habit formation, investment adjustment costs, and variable capital utilization; the Keynesian features include monopolistic competition in goods and labor markets, and nominal price and wage rigidities. The model also includes growth in the form of a non-stationary technology shock, as in Altig, Christiano, Eichenbaum, and Lindé (25). 2.1 Households The economy is populated by a continuum of households, indexed by j [, 1]. Each household consumes final goods, supplies a specific type of labor to intermediate goods firms via employment agencies, saves in one-period nominal government bonds, and accumulates physical capital through investment. It transforms physical capital to effective capital by choosing the capital utilization rate, and then rents the effective capital to intermediate goods firms. Household j chooses consumption C t (j), labor supply L t (j), bond holdings B t (j), the rate of capital utilization ν t, investment I t, and physical capital K t to maximize the intertemporal utility function { E t β s ε b t+s s= [ log (C t+s (j) hc t+s 1 (j)) ε l t+s L t+s (j) 1+ω ] }, (1) 1 + ω where β is a discount factor, h measures the degree of habits in consumption, ω is the inverse Frisch elasticity of labor supply, ε b t is an intertemporal preference shock, and ε l t is a shock to the disutility of supplying labor. The intertemporal preference shock has mean unity and is assumed to follow the autoregressive process log ε b t = ρ b log ε b t 1 + ζ b t, ζ b t i.i.d. N(, σ 2 b ). (2) The labor disutility shock has mean ε l. We will explore different processes for this shock in the estimated model in Section 4 below. The capital utilization rate ν t transforms physical capital Kt into effective capital K t according to K t = ν t Kt 1, (3) and the effective capital is rented to intermediate goods firms at the nominal rental rate R k t. The cost of capital utilization per unit of physical capital is given by A(ν t ), and we assume that ν t = 1 in steady state, A(1) =, and A (1)/A (1) = η ν, as in Christiano, Eichenbaum, 4

9 and Evans (25) and others. Physical capital accumulates according to K t = (1 δ) K t 1 + ε i t [ ( )] It 1 S I t, (4) I t 1 where δ is the depreciation rate of capital, ε i t is a shock to the marginal efficiency of investment that has mean unity, and S( ) is an adjustment cost function which satisfies S (γ z ) = S (γ z ) = and S (γ z ) = η k >, where γ z is the economy s (gross) growth rate in steady state. The investment shock follows the process log ε i t = ρ i log ε i t 1 + ζ i t, ζ i t i.i.d. N(, σ 2 i ). (5) Let P t be the nominal price level, R t the one-period nominal (gross) interest rate, A t (j) the net returns from a portfolio of state-contingent securities, W t the nominal wage, Π t nominal profits from ownership of firms, and T t nominal lump-sum transfers. Household j s budget constraint is then given by P t C t +P t I t +B t = T t +R t 1 B t 1 +A t (j)+π t +W t (j)l t (j)+r k t ν t Kt 1 P t A (ν t ) K t 1. (6) Assuming that households have access to a complete set of state-contingent securities, consumption and asset holdings are the same for all households. The first-order conditions for consumption, bond holdings, investment, physical capital, and effective capital are then given by Λ t = ε b t C t hc t 1 βhe t Λ t = βr t E t { 1 = Q t ε i t P t Λ t+1 P t+1 ( It [ 1 S I t 1 { ε b t+1 }, (7) C t+1 hc t }, (8) ) I ( )] { t S It Λ t+1 + βe t Q t+1 ε i t+1 I t 1 I t 1 Λ t ( It+1 I t ) 2 ( ) } S It+1, (9) { [ ]} Λ t+1 R k Q t = βe t+1 t ν t+1 A (ν t+1 ) + (1 δ)q t+1, (1) Λ t P t+1 R k t = P t A (ν t ), where Λ t is the marginal utility of consumption and Q t is Tobin s Q, that is, the marginal value of capital relative to consumption. 2.2 Final goods producing firms A perfectly competitive sector combines a continuum of intermediate goods Y t (i) indexed by i [, 1] into a final consumption good Y t according to the production function [ 1 Y t = ] ε p t Y t (i) 1/εp t di, (12) I t (11) 5

10 where ε p t is a time-varying measure of substitutability across differentiated intermediate goods. This substitutability implies a time-varying (gross) markup of prices over marginal cost equal to ε p t that is assumed to follow the process log ε p t = ( 1 ρ p ) log ε p + ρ p log ε p t 1 + ζp t, ζp t i.i.d. N(, σ2 p), (13) where ε p is the steady-state price markup. Profit maximization by final goods producing firms yields the set of demand equations [ ] Pt (i) ε p t /(εp t 1) Y t (i) = Y t, (14) P t where P t (i) is the price of intermediate good i and P t is an aggregate price index given by [ 1 P t = ] ε p P t (i) 1/(εp t 1) t 1 di. (15) 2.3 Intermediate goods producing firms Each firm in the intermediate goods sector produces a differentiated intermediate good i using capital and labor inputs according to the production function { } Y t (i) = max K t (i) α [Z t L t (i)] 1 α Z t F,, (16) where α is the capital share, Z t is a labor-augmenting productivity factor, whose growth rate ε z t = Z t /Z t 1 follows a stationary exogenous process with steady-state value ε z which corresponds to the economy s steady-state (gross) growth rate γ z, and F is a fixed cost that ensures that profits are zero in steady state. The rate of technology growth is assumed to follow log ε z t = (1 ρ z ) log ε z + ρ z log ε z t 1 + ζ z t, ζ z t i.i.d. N(, σ 2 z). (17) Thus, technology is non-stationary in levels but stationary in growth rates, following Altig, Christiano, Eichenbaum, and Lindé (25). We assume that capital is perfectly mobile across firms and that there is a competitive rental market for capital. and Cost minimization implies that nominal marginal cost MC t is determined by MC t (i) = W t (1 α)z 1 α t (L t (i)/k t (i)) α (18) MC t (i) = r k t αzt 1 α α 1, (19) (K t (i)/l t (i)) so nominal marginal cost is common across firms and given by MC t = [ α α (1 α) 1 α] 1 (Wt /Z t ) 1 α ( r k t ) α. (2) 6

11 Prices of intermediate goods are set in a staggered fashion, following Calvo (1983). Thus, only a fraction 1 θ p of firms is able to reoptimize their price in any given period. The remaining fraction is assumed to index the price to a combination of past inflation and steady-state inflation according to the rule P t (i) = P t 1 (i)π γ p t 1 π1 γ p, (21) where π t = P t /P t 1 is the gross rate of inflation with steady-state value π and γ p [, 1]. If the indexation parameter γ p is equal to zero, firms index fully to steady-state inflation, as in Yun (1996); if γ p = 1, firms index fully to lagged inflation, as in Christiano, Eichenbaum, and Evans (25). Firms that are able to set their price optimally instead choose their price P t (i) to maximize the present value of future profits over the expected life-time of the price contract: E t { s= (βθ p ) s Λ t+s Λ t { Π t,t+s P t (i)y t+s (i) W t+s L t+s (i) R k t+sk t+s (i)} }, (22) where 1 for s =, Π t,t+s = s k=1 πγ p t+k 1 π1 γ p for s 1., (23) subject to the demand from final goods producing firms in equation (14), As all firms changing their price at time t face the same problem, they all set the same optimal price P t. The first-order condition associated with their maximization problem is E t { s= [ (βθ p ) s Λt+s ( Y t,t+s Πt,t+s Pt ε p t+s Λ MC ) ]} t+s =, (24) t where Y t,t+s is demand facing the firm at time t, given the price P t. In the limiting case of full price flexibility (θ p = ) the optimal price is P t = ε p t MC t, (25) that is, an exogenous markup ε p t over nominal marginal cost. With staggered price setting, the price index P t evolves according to P t = ( ) [(1 θ p ) (P t ) 1/(εpt 1) + θ p π γ 1/(ε p p t 1 π1 γ pp 1)] ε p t 1 t t 1, (26) and the markup over marginal cost is endogenous. 2.4 The labor market As in Erceg, Henderson, and Levin (2), each household is a monopolistic supplier of specialized labor L t (j), which is combined by perfectly competitive employment agencies 7

12 into labor services L t according to [ 1 εw t L t = L t (j) 1/εw t dj], (27) where ε w t is a time-varying measure of substitutability across labor varieties that translates into a time-varying (gross) markup of wages over the marginal rate of substitution between consumption and leisure. The wage markup shock has mean ε w, and as in the case of the labor disutility shock, we will explore different stochastic processes for the wage markup shock below. Profit maximization by employment agencies yields the set of demand equations [ ] Wt (j) ε w t /(ε w t 1) L t (j) = L t, (28) W t for each j, where W t (j) is the wage received from employment agencies by the household supplying labor variety j, and W t is the aggregate wage index given by [ 1 W t = ] ε w t 1 W t (j) 1/(εw t 1) dj. (29) In any given period, a fraction 1 θ w of households is able to set their wage optimally. Similar to the price indexation scheme, the remaining fraction indexes the wage to the steadystate growth rate γ z and a combination of past inflation and steady-state inflation according to W t (j) = W t 1 (j)γ z π γ w t 1 π 1 γ w. (3) The optimizing households choose the wage to maximize { [ E t (βθ w ) s W t (j) Λ t+s L t+s (j) ε b P t+sε l L t+s (j) 1+ω ] } t+s, (31) t+s 1 + ω s= subject to the labor demand equation (28). All optimizing households then set the same optimal wage W t to satisfy the first-order condition { E t (βθ w ) s Λ t+s L t,t+s [Π w Wt t,t+s ε w P t+sε b t+sε l t+s t+s s= L ω t,t+s Λ t+s ] } =, (32) where L t,t+s is labor demand facing the household at time t given the wage W t, and Π w 1 for s =, t,t+s = (33) s k=1 γ zπ γ w t+k 1 π 1 γ w for s 1. The limiting case of full wage flexibility (θ w = ) implies that W t P t = ε w t ε b tε l L ω t t, Λ t (34) 8

13 so the real wage is set as an exogenous markup ε w t over the marginal rate of substitution. With staggered wages, the aggregate wage index W t evolves according to W t = [ (1 θ w ) (W t ) 1/(εw t 1) + θ w ( γz π γ w t 1 π 1 γ ww t 1 ) 1/(ε w t 1) ] ε w t 1, (35) implying an endogenous wage markup. 2.5 Government The government sets public spending G t according to G t = [ 1 1 ] ε g Y t, t (36) where ε g t is a government spending shock that follows the process log ε g t = ( 1 ρ g ) log ε g + ρ g log ε g t 1 + ζg t, ζg t i.i.d. N(, σ2 g). (37) The nominal interest rate R t is set using the monetary policy rule 4 R t R = ( Rt 1 R ) ρs [( ) rπ ( ) πt Yt /Y ry ] 1 ρs t 1 ε r t, (38) π t where π t is a time-varying target for inflation, which follows γ z log π t = (1 ρ ) log π + ρ log π t 1 + ζ t, ζ t i.i.d. N(, σ 2 ). (39) and ε r t is a monetary policy shock which is i.i.d. (in logarithms) with mean unity and variance σ 2 r. Thus the monetary policy rule is affected by two different shock processes: one persistent and one i.i.d. Although we will call the persistent shock an inflation target shock, it could in principle represent any persistent deviation from the monetary policy rule, for instance, errors in the perception of the long-run growth rate γ z. 2.6 Market clearing Finally, to close the model, the resource constraint implies that output is equal to the sum of consumption, investment, government spending, and the capital utilization costs: Y t = C t + I t + G t + A (ν t ) K t 1. (4) 4 We specify the monetary policy rule in terms of output growth rather than the output gap, defined as the deviation of output from its potential level (the level under flexible prices and wages). Thus, we implicitly assume that the central bank either is unable to observe the output gap or is unwilling to let monetary policy depend on its estimate of the output gap. One advantage is that our estimates of the benchmark model are independent of the evolution of the output gap, and therefore of the interpretation of structural shocks, which may be problematic in this class of models (see below). In Section 6 we study a version of the model where the monetary policy rule is specified in terms of the output gap. The quantitative results are largely unchanged relative to the benchmark model. 9

14 2.7 Model summary The complete model consists of 17 endogenous variables determined by 17 equations: the capital utilization equation (3), the capital accumulation equation (4), the households first-order conditions (7) (11), the production function (16), the marginal cost equations (18) (19), the optimal price and wage setting equations (24) and (32), the aggregate price and wage indices (26) and (35), the rules for government spending and monetary policy in (36) and (38), and the resource constraint (4). In addition there are nine exogenous shocks: to households intertemporal preferences ε b t, the disutility of labor ε l t, labor-augmenting technology ε z t, investment ε i t, government spending ε g t, the price and wage markups εp t and εw t, the inflation target π t, and monetary policy ε r t. Output, the capital stock, investment, consumption, government spending, and the real wage all share the common stochastic trend introduced by the non-stationary technology shock ε z t. Therefore, the model is rewritten in stationary form by normalizing these variables by the non-stationary technology shock, and then log-linearized around its steady state. The stationary model, the steady state, and the log-linearized model are described in Appendix A. 5 3 The output gap and the labor wedge in the theoretical model Below we will estimate our model and use it to study the evolution of the output gap and the labor wedge. In this section we use the theoretical model to define these concepts, we show the relationship between the output gap and the labor wedge, and we discuss how to interpret movements in the labor wedge. 3.1 Efficient and potential output The RBC model at the core of our model economy implies an efficient allocation where competition is perfect and there are no price and wage rigidities. That is, prices and wages are flexible and markups are zero. In this allocation all variables are at their efficient levels and fluctuate over time as agents respond efficiently to structural disturbances to technology and preferences. This hypothetical economy is affected by five of our nine shocks: those to technology, investment, intertemporal preferences, the disutility of labor, and government spending. We will label these efficient shocks. The remaining four shocks to the wage and price markups, monetary policy, and the inflation target instead appear only in the model with sticky prices and wages and time-varying markups. We therefore label these as inefficient shocks. Due to the presence of imperfect competition (and thus positive average price and wage markups), the steady-state level of output in the model with sticky prices and wages is lower than in the efficient allocation. An alternative allocation that has the same steady-state level 5 In addition, we supplement the log-linearized model with a block of equations that determines the allocation with flexible prices and wages. Although this block is not used when estimating the benchmark model (where monetary policy does not respond to the output gap), it is useful when constructing different measures of potential output in Section 5. 1

15 as actual output is the potential level of output, defined as the allocation with flexible prices and wages but imperfect competition and markups held constant at their steady-state levels. 6 The potential level of output is affected by the same shocks as the efficient level, and can be shown to be at a constant distance from the efficient level at all times (see Appendix B). Thus, fluctuations in efficient output are reflected one-for-one in fluctuations in potential output. 7 The focus of our analysis is on this measure of potential output, and we define the output gap as the percent deviation of actual output from potential. (The output gap is therefore zero on average.) Following Woodford (23) and Adolfson, Laséen, Lindé, and Svensson (28), we distinguish between two different measures of potential output. The first is derived from the allocation where prices and wages have been flexible forever, and thus uses the state variables from this allocation. 8 We call this the unconditional potential output. The second measure instead uses the state variables in the allocation with sticky prices and wages. This measure, which we call conditional potential output, is taken from an allocation where prices and wages have been sticky in the past, and then unexpectedly become flexible and are expected to remain flexible in the future. While it is straightforward to derive the behavior of the unconditional potential output by setting price and wage rigidities and inefficient shocks to zero, the conditional potential output is more involved. Appendix C describes how we calculate conditional potential output from the solution of the model. Most of the existing literature focuses on the unconditional measure. 9 We will instead focus on conditional potential output, and report only some results for the unconditional measure. Importantly, although conditional potential output moves more closely with actual output than does unconditional potential output, the two measures are highly correlated. Therefore, the two measures give a similar picture of fluctuations in the output gap, and our results do not depend on which measure of potential output we use. 3.2 The output gap and the labor wedge Over the business cycle, inefficient fluctuations in output should be at least partly due to inefficient fluctuations in the allocation of labor. In the core model with only efficient fluctuations, the labor input is determined by equalizing households marginal rate of substitution between consumption and leisure and firms marginal product of labor. Inefficient fluctuations in labor allocation in our full model can therefore be characterized by deviations from 6 This definition follows Woodford (23). A closely related concept is the natural level of output, which is the allocation with flexible prices and wages, but including exogenous shocks to price and wage markups. See also Justiniano and Primiceri (28). 7 The constant distance between the efficient and potential levels of output is not a manifestation of the divide coincidence discussed by Blanchard and Galí (27), which relates to the distance between the efficient and natural levels of output. The divine coincidence fails in our model for (at least) two reasons: price markup shocks lead to a time-varying wedge between the efficient and natural levels of output, and wage rigidities mean that it is not optimal for monetary policy to simultaneously stabilize the efficient output gap and price inflation even if this had been feasible. 8 In the model with flexible prices and wages, the state variables are the physical stock of capital, lagged consumption, and lagged investment. 9 Examples include Neiss and Nelson (23), (25), Edge, Kiley, and Laforte (28), Sala, Söderström, and Trigari (28), and Justiniano and Primiceri (28). Coenen, Smets, and Vetlov (29) instead focus on the conditional measure. 11

16 this efficiency condition. 1 Such deviations, labeled the labor wedge by Chari, Kehoe, and McGrattan (27), have been studied extensively in the literature (examples include Hall (1997), Chari, Kehoe, and McGrattan (27), Galí, Gertler, and López-Salido (27), and Shimer (29)). Letting x t denote the log deviation of any variable X t from its steady-state level, the log-linearized version of our model implies that the marginal rate of substitution and the marginal product of labor are given by 11 mrs t = ω l t λ t + ε b t + ε l t, (41) mpl t = α k t α l t. (42) The labor wedge is then determined as wedge t mrs t mpl t = (α + ω) l t λ t α k t + ε b t + ε l t, (43) and is a measure of inefficiency in the allocation of labor. To characterize the inefficient fluctuations in output caused by the inefficient allocation of labor, we relate the labor wedge to the output gap. We use the fact that the labor wedge is zero in the log-linearized efficient allocation, as well as in the potential allocation (that differs from the efficient allocation only by a constant). We can then write the wedge in equation (43) in terms of the deviations of hours, the marginal utility of consumption, and effective capital from their levels in the potential allocation (indexed by p ) as ) ( λt λ p t wedge t = (α + ω) ( lt l p t ) α ( kt k p t ). (44) The production function (16) implies that the log-linearized output gap is given by ŷ t ŷ p t = Y + F Y [ α ( kt k ) p t + (1 α) ( lt l )] p t. (45) Combining equations (44) and (45) we can write the output gap in terms of the labor wedge as ŷ t ŷ p t = Y + F Y [ 1 α α + ω wedge t + ( λt λ p t ) α(1 + ω) + ( kt 1 α k ) ] p t. (46) 1 In the efficient allocation, the condition MRS=MPL is obtained through the optimal labor supply choice of households that equalize the real wage and the MRS together with the optimal labor demand choice of firms equalizing the real wage and the MPL. In the context of our model, deviations from this efficiency condition can be generated either by imperfections in the labor market (such as sticky wages) that create a wedge between the MRS and the real wage, or by imperfections in the goods market (such as sticky prices) that drive a wedge between prices and nominal marginal cost, that is, between the real wage and the MPL. Other frictions that are not present in our model may also drive a wedge between the MRS and the MPL and generate an inefficient allocation of labor. For example, Jermann and Quadrini (21) develop a model where financial frictions generate such a wedge. 11 As output, investment, the capital stock, consumption, the marginal utility of consumption, and the real wage are non-stationary in our model, so are the MRS and the MPL. The log-linearized model is defined in terms of the ratios of these variables to the non-stationary technology shock that generates the common stochastic trend. The wedge, however, is stationary in the original model. 12

17 In a simple version of our model, without capital, a government sector, fixed costs in production, and habits in consumption, the output gap is exactly proportional to the wedge: 12 ŷ t ŷ p t = ω wedge t, (47) since λ t λ p t = (ĉ t ĉ p t ) = (ŷ t ŷ p t ) and k t = k p t =. In our more elaborate model, the output gap also depends on the gaps for the marginal utility of consumption and effective capital. 13 We will show below that the output gap is close to proportional to the labor wedge also in our estimated model. us understand movements in the output gap. Therefore, understanding the labor wedge will help In particular, it allows us to understand if movements in the output gap are mainly due to exogenous disturbances or to the endogenous effects of price and wage rigidities. Other authors for instance, Hall (1997) and Shimer (29) have studied closely related variants of our labor wedge. Their interest, however, mainly focused on understanding the fundamental driving forces of business cycle fluctuations, as opposed to the efficiency of those fluctuations. To relate to that literature in the context of our more elaborate model, it is useful to write the labor wedge as the sum of a fundamental component and two unobservable shocks: wedge t = wedge t + ε b t + ε l t, (48) where the fundamental component is closely related to the labor wedge studied by Hall (1997) and Shimer (29), and is given by 14 wedge t = (α + ω) l t λ t α k t. (49) In the simple version of our model, without capital, a government sector, fixed costs in production, and habits in consumption, the fundamental wedge is exactly proportional to hours worked: wedge t = (1 + ω) l t, (5) since λ t = ĉ t = ŷ t = (1 α) l t and k t =. Again, we will show below that the fundamental wedge is roughly proportional to hours also in our estimated model. This strong relationship between the fundamental wedge and hours implies that by understanding the behavior of the fundamental wedge we may gain insights into the determinants of the labor input. This is the sense in which the fundamental wedge is a measure of the driving force of 12 This is also demonstrated by Erceg, Henderson, and Levin (2) and Galí, Gertler, and López-Salido (23) 13 The effective capital gap is composed of a capital utilization gap and a gap for the physical capital stock: kt k p t = ( ν t ν p t ) + ( kt 1 kp t 1). Using the conditional potential allocation, the latter gap is zero, so the effective capital gap is entirely due to capital utilization. 14 Our fundamental wedge differs from the labor wedge in Hall (1997) and Shimer (29) because of habits in consumption and fixed costs in production. 13

18 business cycle fluctuations, as opposed to a measure of the inefficiency of those fluctuations Interpreting the labor wedge To understand and interpret fluctuations in the labor wedge, hours, and the output gap it is useful to decompose the fundamental wedge in equation (49) along two dimensions: an efficient versus an inefficient component, and an endogenous versus an exogenous component. Imperfect competition in labor and product markets implies that the real wage is set as a markup over the marginal rate of substitution, and prices as a markup over nominal marginal cost, given by the nominal wage adjusted for the marginal product of labor. By adding and subtracting the real wage from the expression for the labor wedge, we can write the wedge in terms of the wage and price markups (denoted µ w t and µ p t ):16 wedge t = mrs t mpl t = ( mrs t ŵ t ) + (ŵ t mpl ) t = ( µ w t + µ p t ). (51) Each markup can be decomposed into an endogenous component (due to price and wage rigidities) and an exogenous component (equal to the markup shock, or the desired markup in the absence of price and wage rigidities) as µ w t = µ w t + ε w t, µ p t = µ p t + εp t, (52) (53) where µ w t and µ p t are the endogenous markups. Using this markup decomposition in the definition of the fundamental wedge in equation (48), we can write Efficient Inefficient {}}{{ ( }} ){ wedge t = ( µ w t + µ p t ) ( εw t + ε p t ) ε b t + ε l t ) = ( µ w t + µ p t }{{} ) ( ε w t + ε p t ( ε ) b t + ε l t }{{} Endogenous Exogenous (54) In one dimension, the fundamental wedge is decomposed into an inefficient component (due to the total markups) and an efficient component (due to preference shocks). In another dimension, it is decomposed into an endogenous component (due to price and wage rigidities) and an exogenous component (due to disturbances). The inefficient component is, of course, the total labor wedge in equation (43), and is in part endogenous and in part exogenous. The efficient component is entirely exogenous and is due to intertemporal preference and labor disutility shocks. The endogenous component is entirely inefficient and is due to wage 15 Galí, Gertler, and López-Salido (27) also study the labor wedge in a model similar to Hall (1997) and Shimer (29), but focus on the wedge as a measure of inefficiency. 16 See also Galí, Gertler, and López-Salido (27). In our estimated model, the price markup is typically small (as the real wage moves closely with the marginal product of labor). Thus, the labor wedge is largely determined by the wage markup. 14

19 and price rigidities, whereas the exogenous component includes both efficient and inefficient shocks. Part of the literature has focused on explaining fluctuations in the fundamental wedge. The decomposition in equation (54) shows that these fluctuations can be due to movements in exogenous preference shocks, endogenous markups, or exogenous markup shocks. estimated model will give a complete description of the data and therefore all variables in the model, including the fundamental wedge. It is therefore well suited to interpret movements in the labor wedge, hours, and the output gap. Finally, note that the fundamental wedge itself and its decomposition into the endogenous and exogenous components are independent of whether volatility is generated by efficient shocks to households preferences or inefficient shocks to the price or wage markups. This is useful because it is not always easy to identify and interpret the shocks in the estimated model, as we discuss next. 3.4 Interpreting the structural shocks In some cases, the structural interpretation of the estimated shocks is straightforward: shocks to technology are clearly efficient, whereas shocks to monetary policy do not affect the efficient allocation, and are therefore inefficient. In other cases, the interpretation is less clearcut. It is well known that shocks to the disutility from supplying labor (ε l t in our model) and shocks to the wage markup (ε w t ) are observationally equivalent in the log-linearized version of this class of models. 17 These two shocks enter only in the equation relating the real wage to the marginal rate of substitution. In the log-linearized model this equation is given by ŵ t = γ b [ŵ t 1 π t + γ w π t 1 ε z t ] + γ f [ŵt+1 + π t+1 γ w π t + ε z ] t+1 +γ o [ω l t λ ] t + ε b t + ε l t + ε w t, (55) where γ b, γ f, and γ o are convolutions of the structural parameters (see Appendix A). The real wage is driven by movements in the marginal rate of substitution, given by ω l t λ t + ε b t + ε l t, adjusted for shocks to the wage markup, ε w t. Our Thus, in the log-linearized model the two shocks ε l t and ε w t are observationally equivalent, and if both shocks are present they can only be separately identified if they are assumed to follow different stochastic processes. 18 For our estimated model, the exact interpretation of these two shocks is not important: each shock could be interpreted either as a labor disutility shock or as a wage markup shock. For questions about efficiency and inefficiency, however, as well as normative issues, the interpretation of the two shocks comes to the forefront: it crucially affects the estimates of the output gap and the labor wedge, as these reflect inefficient fluctuations in output and inefficiencies in the allocation of labor. It also affects the decomposition of the fundamental wedge into its efficient and inefficient components. We will explore below how the interpre- 17 Similar issues of interpretation also apply to the price markup shock, which can also be interpreted as an efficient relative-price shock to a flexible-price sector in a two-sector model; see de Walque, Smets, and Wouters (26). The price markup shock is small in our estimated model, however, so its interpretation is quantitatively less important. 18 The intertemporal preference shock ε b t also enters the consumption Euler equation, and can therefore be identified separately from ε l t and ε w t. 15

20 tation of these two shocks affects our estimates of the output gap and the labor wedge and the interpretation of macroeconomic fluctuations. 4 Estimation 4.1 Data and estimation technique We estimate the log-linearized version of the model using quarterly U.S. data from 196Q1 to 29Q2 for seven variables: (1) output growth: the quarterly growth rate of per capita real GDP; (2) investment growth: the quarterly growth rate of per capita real private investment plus real personal consumption expenditures of durable goods; (3) consumption growth: the quarterly growth rate of per capita real personal consumption expenditures of services and nondurable goods; (4) real wage growth: the quarterly growth rate of real compensation per hour; (5) hours worked: hours of all persons divided by population; (6) inflation: the quarterly growth rate of the GDP deflator; and (7) the nominal interest rate: the quarterly average of the federal funds rate. Many of our results will be driven by the behavior of hours over the business cycle. We use data on hours from Francis and Ramey (29) that refer to the total economy (rather than the non-farm business sector) and are adjusted for low-frequency movements due to changes in demographics. These data therefore display less low-frequency behavior than unadjusted data. Data definitions and sources are available in Appendix D. We estimate the model using Bayesian likelihood-based methods (see An and Schorfheide (27) for an overview). Letting θ denote the vector of structural parameters to be estimated and Y the data sample, we use the Kalman filter to calculate the likelihood L(θ, Y), and then combine the likelihood function with a prior distribution of the parameters to be estimated, p(θ), to obtain the posterior distribution, L(θ, Y)p(θ). We use numerical routines to maximize the value of the posterior, and then generate draws from the posterior distribution using the Random-Walk Metropolis-Hastings algorithm. We use growth rates for the non-stationary variables in our data set (output, consumption, investment, and the real wage, which are non-stationary also in the theoretical model) and we write the measurement equation of the Kalman filter to match the seven observable series with their model counterparts. Thus, the state-space form of the model is characterized by the state equation X t = A(θ)X t 1 + B(θ)ζ t, ζ t i.i.d. N(, Σ ζ ), (56) where X t is a vector of endogenous variables, ζ t is a vector of innovations, and θ is a vector of parameters; and the measurement equation Y t = C(θ) + DX t + η t, η t i.i.d. N(, Σ η ), (57) where Y t is a vector of observable variables, that is, Y t = 1 [ log Y t, log I t, log C t, log W t, log L t, log π t, log R t ], (58) 16